The coefficient of x^4 in (2−4x+3x^2 )^(−2) i s ?


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Question Number 54024 by sandip123 last updated on 28/Jan/19

$The coefficient of x^{4} in {(2 - 4 x + {3 x}^{2})}^{- 2} i s ?$
	Answered by tanmay.chaudhury50@gmail.com last updated on 28/Jan/19
	$(2−4x+3x^2 )^(−2) ={2−(4x−3x^2 )}^(−2) =2^(−2) {1−(((4x−3x^2 )/2))}^(−2) now using formula (1−y)^(−2) =1+2y+3y^2 +4y^3 +...+(r+1)x^r +... =2^(−2) [1+2(((4x−3x^2 )/2))+3(((4x−3x^2 )/2))^2 +4(((4x−3x^2 )/2))^3 +5(((4x−3x^2 )/2))^4 +... =2^(−2) [1+4x−3x^2 +(3/4)(4x^2 −24x^3 +9x^4 )+(4/8)(64x^3 −3×16x^2 ×3x^2 +3×4x×9x^4 −27x^6 )+(5/2^4 )(4^4 x^4 +others ignored) ignoring red marked... =2^(−2) [(3/4)×9x^4 +(4/8)×−9×16×x^4 +(5/2^4 )×4^4 x^4 ] =x^4 [((3×9)/(4×2^2 ))−((4×9×16)/(2^2 ×8))+((5×4^4 )/(2^4 ×2^2 ))] =x^4 [((27)/(16))−((18)/1)+((5×2^8 )/2^6 )] =x^4 [((27)/(16))−((18)/1)+((5×2^2 )/1)] =x^4 [((27)/(26))+2] so coefficient=((79)/(26)) pls check steps...$
	${(2 - 4 x + 3 x^{2})}^{- 2}$ $= {2 - (4 x - 3 x^{2})}^{- 2}$ $= 2^{- 2} {1 - (\frac{4 x - 3 x^{2}}{2})}^{- 2}$ $n o w u s i n g f o r m u l a$ ${(1 - y)}^{- 2} = 1 + 2 y + 3 y^{2} + 4 y^{3} + . . . + (r + 1) x^{r} + . . .$ $= 2^{- 2} [1 + 2 (\frac{4 x - 3 x^{2}}{2}) + 3 {(\frac{4 x - 3 x^{2}}{2})}^{2} + 4 {(\frac{4 x - 3 x^{2}}{2})}^{3} + 5 {(\frac{4 x - 3 x^{2}}{2})}^{4} + . . .$ $= 2^{- 2} [1 + 4 x - 3 x^{2} + \frac{3}{4} (4 x^{2} - 24 x^{3} + 9 x^{4}) + \frac{4}{8} (64 x^{3} - 3 \times 16 x^{2} \times 3 x^{2} + 3 \times 4 x \times 9 x^{4} - 27 x^{6}) + \frac{5}{2^{4}} (4^{4} x^{4} + o t h e r s i g n o r e d)$ $i g n o r i n g r e d m a r k e d . . .$ $= 2^{- 2} [\frac{3}{4} \times 9 x^{4} + \frac{4}{8} \times - 9 \times 16 \times x^{4} + \frac{5}{2^{4}} \times 4^{4} x^{4}]$ $= x^{4} [\frac{3 \times 9}{4 \times 2^{2}} - \frac{4 \times 9 \times 16}{2^{2} \times 8} + \frac{5 \times 4^{4}}{2^{4} \times 2^{2}}]$ $= x^{4} [\frac{27}{16} - \frac{18}{1} + \frac{5 \times 2^{8}}{2^{6}}]$ $= x^{4} [\frac{27}{16} - \frac{18}{1} + \frac{5 \times 2^{2}}{1}]$ $= x^{4} [\frac{27}{26} + 2] s o c o e f f i c i e n t = \frac{79}{26}$ $p l s c h e c k s t e p s . . .$
	Commented by sandip123 last updated on 01/Feb/19

	$you will get right answer = 59 / 16$ $error is only in last line . thanks!$
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